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\title{\vspace{-4cm}\textbf{河北师范大学数学分析真题}}
\author{宁鑫雨}
\date{\today}
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\begin{document}
\date{}
\section*{2015年数学分析}
\begin{problem}[本题30分,每题10分]\\
1)求极限$\displaystyle\lim_{x\to 0}\left\{\lim_{n\to\infty}\left[\cos\frac{x}{2}\cos\frac{x}{4}\cdots\cdots\cos\frac{x}{2^n}\right]\right\}$\\
2)设$f(x)=
\left\{
\begin{array}{cc}
    x e^{x^{2}},&-\frac{1}{2}<x<\frac{1}{2};\\ 
    -1,&x\leq-\frac{1}{2}\\ 
\end{array}
\right.
$.
求$\displaystyle\int_{\frac{1}{2}}^{2}f(x-\frac{3}{2})\d x$\\
3)计算二重积分$\iint\limits_{D}|y-x|d x d y$,其中$D$为区域$|x|\leq 1$,$0\leq y \leq 2$.\\
\end{problem}

\begin{problem}[本题10分]
设$\displaystyle\lim_{n\to\infty}a_{n}=0$,$\displaystyle x_{n}=\frac{2a_1+2^2a_2+\cdots+2^na_n}{2+2^{2}+\cdots+2^{n}}$,
用$\varepsilon-N$定义证明$\displaystyle\lim_{x\to\infty}x_{n}=0$\\
\end{problem}

\begin{problem}[本题10分]
设$f(x),f^{\prime}(x)$在$[a,b]$上连续,$f^{\prime\prime}(x)$在$(a,b)$内存在，
且$f(a)=f(b)=0$,在$(a,b)$内存在$c\in(a,b)$,使得$f(c)<0$,
证明：在$(a,b)$内存在$\xi$,使得$f^{\prime\prime}(\xi)>0$
\end{problem}

\begin{problem}[本题10分]
    设$f(x)$在$[0,1]$连续,在$(0,1)$内可导,且满足$\displaystyle f(0)=2015\int_{0}^{\frac{1}{2015}}{}e^{-x^{2}}f(x)d x$\\
    证明：至少存在一点$\xi \in (0,1)$,使得$f^{\prime}(\xi)=2\xi f(\xi)$\\
\end{problem}

\begin{problem}[本题15分]
设$f(x)$在$\left(-\infty,+\infty\right)$上连续,且$\displaystyle\lim_{x\to+\infty}f(x)=A,\displaystyle\lim_{x\to-\infty}f(x)=B$,($A,B$均为有限数)\\
证明：\\
1)$f(x)$在$\left(-\infty,+\infty\right)$上有界\\
2)$f(x)$在$\left(-\infty,+\infty\right)$上一致连续
\end{problem}

\begin{problem}[本题15分]
    设$f(x)$在$\left(-\infty,+\infty\right)$上二阶连续可导,且$f(0)=0$,若$g(x)=\left\{\begin{matrix}\displaystyle \frac{f(x)}{x},x\neq0\\ f^{\prime}(0),x=0\end{matrix}\right.$\\
    证明：\\
    1)$g(x)$在$\left(-\infty,+\infty\right)$上连续;\\
    2)$g(x)$在$\left(-\infty,+\infty\right)$上可微;\\
    3)$g^{\prime}(x)$在$\left(-\infty,+\infty\right)$上连续\\
\end{problem}

\begin{problem}[本题15分]
    设$f(x)$在$[0,2]$上连续,在$(0,2)$上二阶可导,且$\displaystyle\lim_{x\to1}\frac{f(x)}{x-1}=0$,$\displaystyle\int_{1}^{2}f(x)d x=f(2)$,证明：$\exists\xi\in(0,2)$,使得$f^{{\prime}{\prime}}(\xi)=0$\\
\end{problem}

\begin{problem}[本题15分]
    设$f(x)$在$\left[1,+\infty\right)$上单调递增,且无穷积分$\displaystyle\int_{1}^{+\infty}f(x)d x$收敛，证明$\lim\limits_{x\to+\infty}x f(x)=0$\\
\end{problem}

\begin{problem}[本题15分]
    设$a_{0}=3$,对任意$n\geq1$,$\displaystyle na_{n} =\frac{2}{3}a_{n-1}-(n-1)a_{n-1}$,证明:
    当$|x|<1$时，幂级数$\displaystyle\sum_{n=1}^{n}a_{n}x^{n}$收敛,并求其和函数.\\
\end{problem}

\begin{problem}[本题15分]
    设$f(x,y)=x y g(x,y)$,其中$g(x,y)$在$(0,0)$某邻域内连续且$g(0,0)=0$,证明$f(x,y)$在$(0,0)$处可微.\\
\end{problem}
\end{document}